In Section 5 we prove the main result of the paper, Theorem REF .
This result says that if \(P\rightarrow M\)  is a principal bundle in \(s{K}/B\)  such that
\(M\)  is suitably cofibrant and each fiberwise bundle \(P_n\rightarrow M_n\)  is numerable,
then the fiberwise geometric realization \(|P|\rightarrow |M|\)  is a fiberwise principal
bundle in \({K}/B\) . Our method of proof is an adaptation of the arguments
from [1]}, [2]}, [3]}, the main idea being to construct
local sections of \(|P|\rightarrow |M|\)  via a filtration of \(|P|\)  by equivariant fiberwise
neighborhood deformation retracts.
